This work compares the enstrophy with the scalar dissipation rate, as well as the passive scalar variance with the turbulent kinetic energy, in the presence of coherent Kármán vortices in the intermediate wake of a circular cylinder. Measurements are made at $x/d=10$, 20 and 40, where $x$ is the streamwise distance from the cylinder axis and $d$ is the cylinder diameter, with a Reynolds number of $2.5\times 10^{3}$ based on the cylinder diameter and the free-stream velocity. A probe consisting of eight hot wires (four X-wires) and four cold wires is used to measure simultaneously the three components of the fluctuating velocity and vorticity vectors, as well as the fluctuating temperature gradient vector at nominally the same point in the plane of the mean shear. It is found that the enstrophy and scalar dissipation spectra collapse approximately at all wavenumbers except around the Kármán vortex street wavenumber for $x/d\geqslant 20$. The spectral similarity between the streamwise velocity fluctuation $u$ and the passive scalar $\unicode[STIX]{x1D703}$ is better than that between the velocity fluctuation vector $\boldsymbol{q}$ and $\unicode[STIX]{x1D703}$. This is closely related to the highly organized lateral velocity fluctuation $v$ in this flow. The present observations are fully consistent with the expectation that small scales of the velocity and temperature fields are more likely to exhibit a close relationship than scales associated with the bulk of the turbulent energy or scalar variance. The variation across the wake of the time scale ratio between scalar and velocity fields is significantly smaller than that of the turbulent Prandtl number.

We analyse the motion of a current of water migrating under gravity into a hot vapour-saturated porous rock accounting for the vaporization which occurs as the water invades the hot rock. We present a series of similarity solutions to describe the rate of advance of both planar and axisymmetric currents when the total mass of water injected after time t is proportional to tγ. Three distinct cases arise. When γ>1/2 (planar) or γ>1 (axisymmetric), the depth of the current increases at all points from the source, and therefore vaporization occurs at all points on its surface. This case is described by a simple extension of the well-known similarity solutions for non-vaporizing currents. When 0<γ<1/2 (planar) or 0<γ<1 (axisymmetric), there is a region near the source where the depth of the current decreases. The depth only increases at the more distant points. Vaporization therefore only occurs in the leading part of the current where it is advancing into the superheated rock. In this case, we develop modified similarity solutions which account for the vaporization in the distal part of the current. The third case involves the finite release of fluid. Owing to the vaporization, the mass of the current decreases with time. Since there is no injection, the rate of advance of the current can no longer be found by comparing the exponents of time in the local and global equations for mass conservation. Instead, the motion is described by a class of similarity solutions of the second kind, analogous to those described by Barenblatt (1997), in which the total mass of the current is proportional tγ, where γ is a function of the mass fraction which vaporizes, [Fscr ], such that γ→0 as [Fscr ]→0 and γ→−1 as [Fscr ]→1. The model is extended to include the effects of capillary retention of fluid in the pore spaces and we discuss the relevance of our results to the process of liquid reinjection in the geothermal power industry.

The influence of an external laminar flow on the spreading of a viscous gravity current moving over a horizontal floor is studied theoretically and experimentally. The viscous stress exerted by the ambient flow drives the viscous gravity current streamwise with a velocity proportional to the local height of the current. The one-way coupling between the ambient flow and the spread of the current is examined. Similarity and numerical solutions are developed to describe viscous gravity currents spreading from line and point sources. An experimental study of the spreading of viscous gravity currents issuing from a point source in a channel flow, for both constant-flux and instantaneous releases, confirms the essential character of this description.

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.

We consider strongly nonlinear waves in fluids in a uniform gravity field, and demonstrate that an incompressible wave motion, in which pressure remains constant in each fluid parcel, is supported by compressible fluids with free and rigid boundaries. We present exact analytic solutions of nonlinear hydrodynamics equations which describe the incompressible wave motion. The solutions provide an extension of the Gerstner wave in an incompressible fluid with a free boundary to waves in compressible three-dimensionally inhomogeneous moving fluids such as oceans and planetary atmospheres.

A highly rarefied gas, initially in equilibrium, is compressed by an infinite plane piston. The resulting flow is considered when the gas is unbounded and also when the gas is bounded by a second stationary wall. In the first case, the density ahead of the piston is analysed for specularly and diffusely reflecting surfaces. In the second case, the average temperature of the gas is found as a function of the wall separation for specularly reflecting surfaces. This (non-equilibrium) temperature is compared with that generated in the corresponding continuum flow which involves multiple shock wave reflexion. It is shown that, for a given average density gradient, the free molecule temperatures are very much higher than the continuum temperatures.

The dissolution of a multicomponent nanodrop in a sparingly miscible liquid is studied by molecular dynamics (MD) simulations. We studied both binary and ternary systems, in which nanodroplets are formed from one and two components, respectively. Whereas for a single-component droplet the dissolution can easily be calculated, the situation is more complicated for a multicomponent drop, as the interface concentrations of the drop constituents depend on the drop composition, which changes with time. In this study, the variation of the interface concentration with the drop composition is determined from independent ‘numerical experiments’, which are then used in the theoretical model for the dissolution dynamics of a multicomponent drop. The MD simulations reveal that when the interaction strengths between the drop constituents and the surrounding bulk liquid are significantly different, the concentration of the more soluble component near the drop interface may become larger than in the drop bulk. This effect is the larger the smaller the drop radius. While the present study is limited to binary and ternary systems, the same method can be easily extended to a larger number of components.

The gas-flow pattern that occurs when a bubble grows during its passage through a fluidized bed is investigated by applying Davidson's (1961) model in an unsteady manner. It is found that the form of the cloud perceived by an observer of this unsteady motion is significantly different from that predicted by the steady theory. Growth is shown to lead to a reduction in the penetration of the cloud ahead of the bubble and a reduction in the area of overlap between cloud and particles. The cloud boundary also interaepts the bubble boundary, and the gas entering the bubble to sustain its growth does so through its base. There is good agreement with experimental observation of clouds.

The high-Reynolds-number stability of unsteady pipe flow to axisymmetric disturbances is studied using asymptotic analysis. It is shown that as the disturbance amplitude is increased, nonlinear effects first become significant within the critical layer, which moves away from the pipe wall as a result. It is found that the flow stabilizes once the basic profile has become sufficiently fully developed. By tracing the nonlinear neutral curve back to earlier times, it is found that in addition to the wall mode, which arises from a classical upper branch linear stability analysis, there also exists a nonlinear neutral centre mode, governed primarily by inviscid dynamics. The centre mode problem is solved numerically and the results show the existence of a concentrated region of vorticity centred on or close to the pipe axis and propagating downstream at almost the maximum fluid velocity. The connection between this structure and the puffs and slugs of vorticity observed in experiments is discussed.

Wave diffraction, wave forces and wave drift damping due to a floating body performing a slow rotation about the vertical axis (yaw) is considered. The rotation angle of the body may be arbitrary. The angular velocity is assumed small compared to the wave frequency, however. The problem is formulated in the frame of reference following the slow rotation of the body, accounting for non-Newtonian forces. By applying the method of multiple timescales, the fluid flow is determined consistently to leading order in the slow angular velocity and to second order in the wave amplitude. Mathematical solution of the problem is obtained by means of integral equations that are applicable to geometries of arbitrary shape. The wave loads are found by applying conservation of linear and angular momentum. The wave drift damping is expressed by the far-field amplitudes of the wave field and the dipole moments of the time-averaged second-order potential. Numerical results are presented for a ship and a vertical cylinder describing a circular path in the horizontal plane. The results show that the wave drift damping due to a slow yaw motion of a floating body is one order of magnitude larger than the time-averaged forces and moment when there is no rotation. Wave drift damping due to slow rotation and slow translation are found to be of equal importance.

To investigate turbulent properties and the developing mechanisms of the tidally induced bottom boundary layer in the linearly stratified ocean, numerical experiments have been executed with a non-hydrostatic three-dimensional model in the rotating frame, changing the temporal Rossby number Rot = |σ/f|, i.e. the ratio of the tidal frequency σ to the Coriolis parameter f. After the flow transitions to turbulence, the entire water column can be characterized by three layers: the mixed layer where density is homogenized and the flow is turbulent (z < zm); the stratified layer where the initial stratification remains and the flow is laminar (z > zt); and the interfacial layer between them where the flow is turbulent but the stratification remains (zm < z < zt). Turbulence is scaled by the frictional velocity uτ and the mixed-layer thickness zm (uτ and uτ/N where N is the buoyancy frequency) in the mixed (interfacial) layer, and has similarity. The mixed layer is thickened by the process where light water of the upper stratified layer is mixed with the lower unstratified layer water through the interfacial layer. As Rot approaches unity, i.e. near the critical latitude, the mixed layer develops more rapidly according to the following mechanism. As becomes Rot closer to unity, the current shear in the interfacial layer is intensified, since the difference of velocity becomes larger between the lower turbulent mixed and upper laminar stratified layers, and this leads to thickening of the interfacial layer. As a result, density deviation of the water entrained from above becomes larger, and this causes more rapid development of the mixed layer. In terms of the energy conversion from the eddy kinetic energy (EKE) to the potential energy (PE), the efficiency factor β which is the ratio of the conversion rate from EKE to PE to that from the tidal shear to EKE increased from 0.25% for Rot = 0.5 to 3.5% for Rot = 1.05 on average. When the time is normalized by the period required for the mixed layer to be thickened to the unstratified turbulent boundary layer δ = uτ/|f+σ|, the mixed layer development occurred in a similar manner in all cases. This similarity suggests the possibility of universal formulation for the turbulent tidal mixing under stratification.

We study theoretically and numerically the nonlinear seakeeping problem of a submerged or floating body translating with constant forward speed $U$ parallel to the undisturbed free surface while at the same time undergoing a small oscillatory motion and/or encountering small-amplitude waves at frequency $\unicode[STIX]{x1D714}$ . It is known that at the critical frequency corresponding to $\unicode[STIX]{x1D70F}\equiv \unicode[STIX]{x1D714}U/g=1/4$ , where $g$ is the gravitational acceleration, the classical linear solution is unbounded for a single point source, and the inclusion of third-order free-surface nonlinearity due to cubic self-interactions of waves is necessary to remove the associated singularity. Although it has been shown that the linear solution is in fact bounded for a body with full geometry rather than a point source, the solution still varies sharply near the critical frequency. In this work, we show theoretically that for a submerged body, the nonlinear correction to the linear solution due to cubic self-interactions of resonant waves in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ is of first order in the wave steepness (or body motion amplitude), which is the same order as the linear solution. With the inclusion of nonlinear effects in the dispersion relation, the wavenumbers of resonant waves become complex-valued and the resonant waves become evanescent, with their amplitudes vanishing with the distance away from the body. To assist in understanding the theory, we derive the analytic nonlinear solution for the case of a submerged two-dimensional circular cylinder in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ . Independent numerical simulations confirm the analytic solution for the submerged circular cylinder. Finally, we also demonstrate by numerical simulations that similar significant nonlinear effects for a surface-piercing body exist in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ .

We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory.